First of all, I'm sorry I do not have top-notch knowledge to properly formulate the question, so I apologize if some of you do not get the point of my question.

It seems in western music, the de-facto standard for writing music is the staff, and the most fine division of note pitches is the 12-notes system.

Based on this system, we choose subsets of the 12 notes to create scales, and chords to create harmonic movement.

If I want to create an alternative formal system for writing music, would it make sense to just start at the scale level, so instead of referring to notes by their name (C, C#, D, ...) I refer to them by their grade in the scale (I, II, III, ...)? (so, in principle it is not possible to write notes that are not part of the scale, but we could introduce back the notion of accidentals, i.e. IIIb or II#)

What are the advantages and disadvantages of that?

I am aware that a musical piece does not use a fixed musical scale but rather modulates back and forth between different (perhaps similar) scales; this aspect could be as well part of the notation.

Is the standard system more complex than this? (in the sense that it allows more expressivity than this)

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    This is already done with a term called scale degrees. It does not replace standard notation, it's just a different way to view the collection of notes. – Dom Apr 4 '18 at 15:18
  • @Dom I'm aware of that notation, and I also used it in my example, however as you said it does not replace the standard notation; it is mostly used when analyzing melody and chords; sometimes it is used in teaching music theory mostly for chords, but I've never seen it utilized for representing notes of a melodic line. – fferri Apr 4 '18 at 15:25
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    I mean it is used for analysis only because the reduction takes away a lot of information. There's also numbered notation that fits the bill. – Dom Apr 4 '18 at 15:51

Comparing your system to the system of note names, your system has some advantages:

  • you sometimes avoid the need to state a sharp or flat alongside the note name

  • you're including a basic level of analysis (of what scale degree you're playing) in the notation.

It also has some disadvantages:

  • Calling notes 'I, II, III' or '1,2,3' is that assumes that a piece of music stays in a single key. As you say, many pieces of music modulate to other keys, so (again, as you say in the question) you'd need to work out what you do in that situation - do you (for example) put in a marker to denote the key change, and make 'I, II, III' suddenly refer to different notes than they did before the marker?
  • If you habitually refer to notes by number (relative to a scale), what do you say if you want to refer to a particular note outside of the context of a key? You'd have to either still use note names (A, A#, B), or invent yet another notation.

It also seems to me that standard notation (dots on a stave) already gives us one of the advantages of your system, as the key signature already 'tidies away' the need to write accidentals next to any note that's diatonic to the key.

One advantage of your system that would remain would be to give us information about the scale degree - but that's probably something that most musicians find very easy to work out anyway when they know what key they're in.

One aspect that's harder to call as an advantage or disadvantage is the performer's mental model of how the notes they're shown relate to positions on the instrument. The performer (presented with your notation) would always have to mentally map those scale degrees to a particular key on their particular instrument, but then that that way of thinking could make transposing on the fly a lot easier. (Personally, this is actually closer to how I think about music - I hardly ever think in terms of note names).

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    Good answer. Also, calling notes by numbers also assumes every instrument in the orchestra knows what note #1 is. How does this system inform musicians what "key" to play in? – scott Apr 4 '18 at 16:45
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    @scott true - not only would we need to indicate any key changes, we'd need to indicate the initial key too. – topo Reinstate Monica Apr 4 '18 at 17:00
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    Personally, this is actually closer to how I think about music - I hardly ever think in terms of note names - For years I worked that way. A couple of years ago I went to a working pro for some jazz lessons and he said he always thinks in terms of notes, not numbers. I started doing it that way and I like better because particular note names and keys seem to have 'personalities' - but that's an entirely subjective. I also began to see the notes on the staff as I play, even if I'm not looking at a sheet. I find that very beneficial. – Stinkfoot Apr 4 '18 at 20:04
  • @Stinkfoot in a way I'm jealous of people who think of the note names - it seems more friendly (almost romantic :) but I'm really not sure my mind can learn another way to work! – topo Reinstate Monica Apr 4 '18 at 20:19
  • @topomorto - took me about 6 months - I can do it well up to the 7th or 8th fret on a bass (I rarely play above there) . I practiced scales from my head and spelled out every note out loud when I played it and made sure I knew exactly where it was and why it was that note. After a while I started thinking that way when playing. – Stinkfoot Apr 4 '18 at 20:37

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